cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A266329 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(x) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 3, 12, 62, 395, 2994, 26331, 263729, 2964845, 36975858, 506687604, 7568226163, 122388728056, 2130425343621, 39718373337525, 789613850257051, 16674806980716514, 372771700023167862, 8794945626017009781, 218392778569695964100, 5693513850197410142081, 155482323312112362743373, 4438621019461797437443233, 132210153223378852014571364, 4101859859297789141335079684, 132343983668857026899533814277
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2016

Keywords

Comments

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).
What is Limit (a(n)/n!)^(1/n) ? Example: (a(300)/300!)^(1/300) = 1.2409703...
Limit (a(n)/n!)^(1/n) = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.24008610649849766623949... - Vaclav Kotesovec, Aug 21 2017

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 62*x^4/4! + 395*x^5/5! + 2994*x^6/6! + 26331*x^7/7! + 263729*x^8/8! + 2964845*x^9/9! + 36975858*x^10/10! +...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + 2*x + 5*x^2/2! + 17*x^3/3! + 79*x^4/4! + 474*x^5/5! + 3468*x^6/6! + 29799*x^7/7! + 293528*x^8/8! + 3258373*x^9/9! + 40234231*x^10/10! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) - 1,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) + log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx + x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + x) dx:
Integral 1/(exp(x) + x) dx  =  x - 2*x^2/2! + 7*x^3/3! - 37*x^4/4! + 261*x^5/5! - 2301*x^6/6! + 24343*x^7/7! - 300455*x^8/8! + 4238153*x^9/9! - 67255273*x^10/10! +...+ (-1)^(n-1)*A072597(n-1)*x^n/n! +...
so that A( Integral 1/(exp(x) + x) dx ) = exp(x).

Links

Crossrefs

Cf. A266328, A072597, A289739.

Programs

  • Mathematica
    a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *)
  • PARI
    {a(n) = my(A=1+x,B=1+x); for(i=0,n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( 1 + A ) ) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + log(A(x)) dx ).
(2) A(x) = A'(x)/A(x) - log(A(x)).
(3) log(A(x)) = exp(x) * Integral exp(-x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.2400861064984976662394901721056528110217273471501174317019052800276... - Vaclav Kotesovec, Aug 21 2017

A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2*x) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 4, 20, 126, 972, 8876, 93580, 1119328, 14986944, 222184136, 3614288272, 64022264176, 1226914925840, 25295189791296, 558317369479616, 13136590271813856, 328243850207690432, 8680766764223956416, 242245419192494844096, 7113910552105144027136, 219304957649505551899136, 7081169542830272102170752, 238996807468258679150596352
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2016

Keywords

Comments

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 20*x^3/3! + 126*x^4/4! + 972*x^5/5! + 8876*x^6/6! + 93580*x^7/7! + 1119328*x^8/8! + 14986944*x^9/9! + 222184136*x^10/10! +...
such that log(A(x)) = Integral B(x) dx
where B(x) = 1 + 3*x + 10*x^2/2! + 40*x^3/3! + 206*x^4/4! + 1384*x^5/5! + 11644*x^6/6! + 116868*x^7/7! + 1353064*x^8/8! + 17693072*x^9/9! + 257570280*x^10/10! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) - 2,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) + 2*log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx + 2*x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + 2*x) dx:
Integral 1/(exp(x) + 2*x) dx  =  x - 3*x^2/2! + 17*x^3/3! - 145*x^4/4! + 1649*x^5/5! - 23441*x^6/6! + 399865*x^7/7! - 7957881*x^8/8! + 180997857*x^9/9! - 4631289697*x^10/10! +...
so that A( Integral 1/(exp(x) + 2*x) dx ) = exp(x).

Links

Crossrefs

Cf. A266328, A266329.

Programs

  • Mathematica
    a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ 2^(j - 1) a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *)
  • PARI
    {a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( 2 + A ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + 2*x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + 2*log(A(x)) dx ).
(2) A(x) = A'(x)/A(x) - 2*log(A(x)).
(3) log(A(x)) = exp(2*x) * Integral exp(-2*x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + 2*x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(2*x + exp(x)) dx = 1.4650202775490107369040248583790383461628786237838809798971... - Vaclav Kotesovec, Aug 21 2017

A268170 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 2, 5, 16, 65, 326, 1947, 13410, 104181, 900214, 8566655, 89055224, 1004141647, 12204369138, 159036267519, 2211764983734, 32696763676521, 511987792322430, 8465194670035767, 147370831072230860, 2694506417687396995, 51622643862824956898, 1034153511794063402519, 21621325640846679627146
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2016

Keywords

Comments

Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...+ A268171(n)*x^n/n! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) + exp(x) - 1,
(2) B(x) = A'(x)/A(x),
(3) log(A(x)) = Integral B(x) dx,
(4) log(B(x)) = Integral A(x) dx + 1+x - exp(x).
RELATED SERIES.
log(A(x)) = x + x^2/2! + x^3/3! + 2*x^4/4! + 9*x^5/5! + 46*x^6/6! + 245*x^7/7! + 1474*x^8/8! + 10315*x^9/9! + 82174*x^10/10! + 726591*x^11/11! + 7038632*x^12/12! +...
Let J(x) equal the series reversion of log(A(x)); then
J(x) = x - x^2/2! + 2*x^3/3! - 7*x^4/4! + 31*x^5/5! - 172*x^6/6! + 1155*x^7/7! - 9027*x^8/8! + 80676*x^9/9! - 811727*x^10/10! + 9075333*x^11/11! - 111633356*x^12/12! +...
where A(J(x)) = exp(x).

Crossrefs

Cf. A266328, A266329, A266490, A268171.

Programs

  • PARI
    {a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp(1+x - exp(x +x*O(x^n)) + intformal( A ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.

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